George Bernard Shaw was an Irish playwright who was the first person to be awarded for both the Nobel Prize for Literature and an Oscar. He didn't truly establish his name as a playwright until his 40s, and wrote sixty-three plays during his career. Many of his writings dealt with social problems, education, marriage, religion government and class privilege.
Pygmalion (My Fair Lady is based on this play)
Man and Superman
Oscar Wilde another Irish writer and poet is famously known for his biting wit and especially his personal life. One of the most celebrated talents during the Victorian era, his social satires have remained as some of the most widely performed plays in history. His only novel The Picture of Dorian Gray has been adapted to screen as well.
Importance of Being Earnest
A Woman of No Importance
An Ideal Husband (1999 version here)
Lady Windermere's Fan (1985 version link here)